"lower intensity same frequency" causes stopping voltage to be lowered, thus threshold frequency is lowered. Why is this so? It would be really helpful if someone could add in an equation showing that a change in photon intensity alters threshold frequency.

"lower intensity same frequency" causes stopping voltage to be lowered, thus threshold frequency is lowered. Why is this so? It would be really helpful if someone could add in an equation showing that a change in photon intensity alters threshold frequency.

You've misunderstood the diagram. What is happening there is, the curve comes in at a different angle to the same intercept. So both the green and blue curves have zero current at -3 volts. The red curve crosses at -2 volts. So the stopping voltage is -3 volts for both the reference beam and the lower intensity, same frequency beam.

It's just that pixels are finite size, so you can't really see it correctly on the graph.

This corresponds to the idea that a photon is a bundle of energy. The size of that bundle is proportional to the frequency, E = h nu. A more intense beam of the same frequency is just more bundles of the same size.

It's not clear what you mean by "threshold frequency" since no such term is defined on the graph.

You've misunderstood the diagram. What is happening there is, the curve comes in at a different angle to the same intercept. So both the green and blue curves have zero current at -3 volts. The red curve crosses at -2 volts. So the stopping voltage is -3 volts for both the reference beam and the lower intensity, same frequency beam.

It's just that pixels are finite size, so you can't really see it correctly on the graph.

This corresponds to the idea that a photon is a bundle of energy. The size of that bundle is proportional to the frequency, E = h nu. A more intense beam of the same frequency is just more bundles of the same size.

It's not clear what you mean by "threshold frequency" since no such term is defined on the graph.

Exactly yours (the OP) is not the same graph as is mostly shown to demonstrate the photoelectric effect. (As in this link.)
Also, all three lines on the graph relate to light which is above the threshold frequency.